The Cheeger constant for distance-regular graphs Greg Markowsky - - PowerPoint PPT Presentation

The Cheeger constant for distance-regular graphs

Greg Markowsky (joint work with Jacobus Koolen and Zhi Qiao) August 9, 2019

Graphs A graph is a set of vertices V(can be taken to be {1, 2, . . . , n}) and edges E, where each edge is an element of V × V. We assume all graphs in this talk are simple, which means that (a, a) / ∈ E and E has no repeated elements, and undirected, which means that (a, b) and (b, a) represent the same edge.

Distance-transitive graphs Let d(x, z) denote the length of the shortest path between x and z within G. A graph is distance-transitive if, whenever there are points X1, z1, x2, z2 such that d(x1, z1) = d(x2, z2), there is an automorphism γ of G such that γ(x1) = x2, γ(z1) = z2. An automorphism is a bijection from the vertex set of G to itself, with the property that γ(u) ∼ γ(v) if, and only if, u ∼ v.

Distance-regular graphs A graph is distance-regular if, for any points x and z within G, the sizes of the following sets depends only on d(x, z): B = {v|d(x, v) = d(x, z) − 1}

• {v|d(z, v) = 1}

A = {v|d(x, v) = d(x, z)}

• {v|d(z, v) = 1}

C = {v|d(x, v) = d(x, z) + 1}

• {v|d(z, v) = 1}

If a graph is distance transitive, it is distance regular. If d(x, z) = i, we define bi, ai, and ci to be the sizes of the three sets above.

Distance-regular graphs - Examples Hamming graphs:Let d, q be positive integers. The vertex set is elements of {1, 2, . . . , q}d. Two vertices are adjacent if they differ in exactly one component. This family of graphs includes the hypercubes.

Distance-regular graphs - Examples Hamming graphs:Let d, q be positive integers. The vertex set is elements of {1, 2, . . . , q}d. Two vertices are adjacent if they differ in exactly one component. This family of graphs includes the hypercubes. Johnson graphs: Let k ≤ n be positive integers. The vertex set is all subsets of {1, 2, . . . , n} of size k. Two vertices are adjacent if their intersection has size k − 1.

Distance-regular graphs - Examples Hamming graphs:Let d, q be positive integers. The vertex set is elements of {1, 2, . . . , q}d. Two vertices are adjacent if they differ in exactly one component. This family of graphs includes the hypercubes. Johnson graphs: Let k ≤ n be positive integers. The vertex set is all subsets of {1, 2, . . . , n} of size k. Two vertices are adjacent if their intersection has size k − 1. The Hamming and Johnson graphs are important in coding

• theory. In particular Hamming graphs arise naturally in the

study of error-correcting codes.

Distance-regular graphs - Examples Odd Graphs: Let n be a positive integer. The vertex set is all subsets of {1, 2, . . . , 2n − 1} of size n − 1. Two vertices are adjacent if their intersection is empty.

Distance-regular graphs - Examples Odd Graphs: Let n be a positive integer. The vertex set is all subsets of {1, 2, . . . , 2n − 1} of size n − 1. Two vertices are adjacent if their intersection is empty. Grassman Graphs: Let V be a vector space of dimension n

• ver the finite field F with q elements (q a prime power). The

vertices are the subspaces of dimension t (over F) of V and such two vertices are adjacent if their intersection is a vector space of dimension t − 1. Grassman graphs are relevant to certain questions in quantum physics.

Cheeger constant The Cheeger constant hG of a graph G is a prominent measure

• f the connectivity of G, and is defined as

hG = inf E[S, Sc] vol(S) |S ⊂ V(G) with |S| ≤ |V(G)| 2

• ,

(1) where V(G) is the vertex set of G, vol(S) is the sum of the valencies of the vertices in S, Sc is the complement of S in V(G), |S| is the number of vertices in S, and for any sets A, B we use E[A, B] to denote the number of edges in G which connect a point in A with a point in B.

Graph eigenvalues For a distance-regular graph G of diameter D, we will write k = θ0 > θ1 > . . . > θD to describe the eigenvalues of the adjacency matrix A of G, and refer to θ0, . . . , θD as simply the eigenvalues of G.

Graph eigenvalues For a distance-regular graph G of diameter D, we will write k = θ0 > θ1 > . . . > θD to describe the eigenvalues of the adjacency matrix A of G, and refer to θ0, . . . , θD as simply the eigenvalues of G. The Laplacian matrix (sometimes referred to as the normalized Laplacian) L = I − 1

k A will therefore have

eigenvalues 0 = λ0 < λ1 < . . . < λD, where the relationship λi = k−θi

k

holds.

Graph eigenvalues For a distance-regular graph G of diameter D, we will write k = θ0 > θ1 > . . . > θD to describe the eigenvalues of the adjacency matrix A of G, and refer to θ0, . . . , θD as simply the eigenvalues of G. The Laplacian matrix (sometimes referred to as the normalized Laplacian) L = I − 1

k A will therefore have

eigenvalues 0 = λ0 < λ1 < . . . < λD, where the relationship λi = k−θi

k

• holds. We will refer to λ0, . . . , λD as the Laplacian

eigenvalues of G.

The connection between eigenvalues and the Cheeger constant Theorem Let λ1 be the smallest positive eigenvalue of the Laplacian matrix of G. Then λ1 2 ≤ hG ≤

• λ1(2 − λ1).

(2)

The connection between eigenvalues and the Cheeger constant Theorem Let λ1 be the smallest positive eigenvalue of the Laplacian matrix of G. Then λ1 2 ≤ hG ≤

• λ1(2 − λ1).

(2) The lower bound can be proved by eigenvalue interlacing.

Our conjecture Conjecture Suppose G is a distance-regular graph, and λ1 is the smallest positive eigenvalue of the Laplacian matrix of G. Then λ1 2 ≤ hG ≤ λ1. (3)

Our conjecture Conjecture Suppose G is a distance-regular graph, and λ1 is the smallest positive eigenvalue of the Laplacian matrix of G. Then λ1 2 ≤ hG ≤ λ1. (3) We do not have a complete proof of this, but we have proved it in a number of cases.

The following are the major known families of infinite families of distance-regular graphs. Johnson graphs. Hamming graphs. Doob graphs. Halved n-cubes. Folded n-cubes. Folded halved 2n-cubes. Odd graphs. Doubled odd graphs. Grassmann graphs. Twisted Grassmann graphs. Doubled Grassman graphs. Bilinear forms graphs. Alternating forms graphs. Hermitian forms graphs. Quadratic forms graph Dual polar graphs and Hemmeter graphs. Half dual polar and Ustimenko graphs.

The following are the major known families of infinite families of distance-regular graphs. Johnson graphs. Hamming graphs. Doob graphs. Halved n-cubes. Folded n-cubes. Folded halved 2n-cubes. Odd graphs. Doubled odd graphs. Grassmann graphs. Twisted Grassmann graphs. Doubled Grassman graphs. Bilinear forms graphs. Alternating forms graphs. Hermitian forms graphs. Quadratic forms graph Dual polar graphs and Hemmeter graphs. Half dual polar and Ustimenko graphs. We have proved the conjecture for all of these, except for the doubled Grassman graphs with q = 2 or 3.

Strongly regular graph A distance-regular graph with diameter 2 is called strongly regular.

Strongly regular graph A distance-regular graph with diameter 2 is called strongly

• regular. We have proved the conjecture for all strongly regular

graphs.

Strongly regular graph A distance-regular graph with diameter 2 is called strongly

• regular. We have proved the conjecture for all strongly regular
• graphs. The following simple lemma took care of most of them.

Lemma: If G is a strongly regular graph, then hG ≤ max( b1

k+1, c2 k ).

Graphs with small valency The distance-regular graphs with valencies 3 and 4 are completely classified, and we have verified the conjecture for all

• f them except the following two graphs:

The flag graph of GH(2, 2), with intersection array {4, 2, 2, 2, 2, 2; 1, 1, 1, 1, 1, 2}. Here λ1 = 3−

√ 6 4

≈ .138. The incidence graph of GH(3, 3), with intersection array {4, 3, 3, 3, 3, 3; 1, 1, 1, 1, 1, 4}. Here λ1 = 1

4.

Graphs with small valency The distance-regular graphs with valencies 3 and 4 are completely classified, and we have verified the conjecture for all

• f them except the following two graphs:

The flag graph of GH(2, 2), with intersection array {4, 2, 2, 2, 2, 2; 1, 1, 1, 1, 1, 2}. Here λ1 = 3−

√ 6 4

≈ .138. The incidence graph of GH(3, 3), with intersection array {4, 3, 3, 3, 3, 3; 1, 1, 1, 1, 1, 4}. Here λ1 = 1

4.

The question remains open for these.

Other classes We also proved the conjecture for several other classes of distance-regular graphs. These are Diameter 3 and bipartite or antipodal. Incidence graphs of generalized quadrangles GQ(q, q) with q = 3, 4. Incidence graphs of generalized hexagons GH(q, q) with q = 3, 4, 5, 7, 8, 9.

References F . Chung, Spectral Graph Theory, 1994.

• N. Biggs, Algebraic Graph Theory, 1993.
• A. Brouwer, A. Cohen, and A. Neumaier, Distance-regular

graphs, 1989.

• C. Godsil and G. Royle, Algebraic Graph Theory, 1993.
• J. Koolen, G. Markowsky, Z. Qiao On the Cheeger constant

for distance-regular graphs, arXiv:1103.2810.